"Eat in" is when you eat the food at the same place as you buy it. For example, a restaurant. "Takeaway" or "Take out" is when you eat the food some other place. For example, you buy pizza at a pizza restaurant but take it home to eat it.

Adding to Mojave's correct response, in the U.S., a restaurant employee may ask you if you want your food "For Here?" or "To go?". Plus, "Take-away/Take-out" is used as a pronoun too: "Let's call 'China King' for some take-out", and "I don't feel like cooking tonight so let's just get take-out".